diff --git a/BezierKit/BezierKitTests/AugmentedGraphTests.swift b/BezierKit/BezierKitTests/AugmentedGraphTests.swift
index d7e5ac5e..c154d98b 100644
--- a/BezierKit/BezierKitTests/AugmentedGraphTests.swift
+++ b/BezierKit/BezierKitTests/AugmentedGraphTests.swift
@@ -185,7 +185,7 @@ class AugmentedGraphTests: XCTestCase {
     }
     
 //    func testMultipleIntersectionsSameElement() {
-//        
+//
 //    }
     
 }
diff --git a/BezierKit/BezierKitTests/PathTests.swift b/BezierKit/BezierKitTests/PathTests.swift
index 74da4451..ec9aa4df 100644
--- a/BezierKit/BezierKitTests/PathTests.swift
+++ b/BezierKit/BezierKitTests/PathTests.swift
@@ -7,6 +7,7 @@
 //
 
 import XCTest
+import CoreGraphics
 @testable import BezierKit
 
 class PathTests: XCTestCase {
@@ -116,20 +117,15 @@ class PathTests: XCTestCase {
     }
     
     func testIntersects() {
-
-        // TODO: improved unit tests ... currently this test is very lax and allows duplicated intersections
-        let circleCGPath = CGMutablePath()
-        circleCGPath.addEllipse(in: CGRect(origin: CGPoint(x: 2.0, y: 3.0), size: CGSize(width: 2.0, height: 2.0)))
-        
+        let circleCGPath = CGPath(ellipseIn: CGRect(x: 2.0, y: 3.0, width: 2.0, height: 2.0), transform: nil)
         let circlePath = Path(cgPath: circleCGPath) // a circle centered at (3, 4) with radius 2
         
-        let rectangleCGPath = CGMutablePath()
-        rectangleCGPath.addRect(CGRect(origin: CGPoint(x: 3.0, y: 4.0), size: CGSize(width: 2.0, height: 2.0)))
-        
+        let rectangleCGPath = CGPath(rect: CGRect(x: 3.0, y: 4.0, width: 2.0, height: 2.0), transform: nil)
         let rectanglePath = Path(cgPath: rectangleCGPath)
         
         let intersections = rectanglePath.intersects(path: circlePath).map { rectanglePath.point(at: $0.indexedPathLocation1 ) }
         
+        XCTAssertEqual(intersections.count, 2)
         XCTAssert(intersections.contains(CGPoint(x: 4.0, y: 4.0)))
         XCTAssert(intersections.contains(CGPoint(x: 3.0, y: 5.0)))
     }
@@ -410,6 +406,50 @@ class PathTests: XCTestCase {
         XCTAssertEqual(circle.subtracting(biggerCircle).subpaths.count, 0)
     }
     
+    func testSubtractingEdgeCase1() {
+        // this is a specific edge case test of `subtracting`. There was an issue where if a path element intersected at the exact border between
+        // two elements on the other path it would count as two intersections. The winding count would then be incremented twice on the way in
+        // but only once on the way out. So the entrance would be recognized but the exit not recognized.
+
+        let rectangle = Path(cgPath: CGPath(rect: CGRect(x: -1, y: -1, width: 4, height: 3), transform: nil))
+        let circle    = Path(cgPath: CGPath(ellipseIn: CGRect(x: 0, y: 0, width: 4, height: 4), transform: nil))
+        
+        // the circle intersects the rect at (0,2) and (3, 0.26792) ... the last number being exactly 2 - sqrt(3)
+        let difference = rectangle.subtracting(circle)
+        XCTAssertEqual(difference.subpaths.count, 1)
+        XCTAssertFalse(difference.contains(CGPoint(x: 2.0, y: 2.0)))
+    }
+    
+    func testSubtractingEdgeCase2() {
+        
+        // this unit test demosntrates an issue that came up in development where the logic for the winding direction
+        // when corners intersect was not quite correct.
+        
+        let square1 = Path(cgPath: CGPath(rect: CGRect(x: 0.0, y: 0.0, width: 2.0, height: 2.0), transform: nil))
+        let square2CGPath = CGMutablePath()
+        square2CGPath.move(to: CGPoint.zero)
+        square2CGPath.addLine(to: CGPoint(x: 1.0, y: -1.0))
+        square2CGPath.addLine(to: CGPoint(x: 2.0, y: 0.0))
+        square2CGPath.addLine(to: CGPoint(x: 1.0, y: 1.0))
+        square2CGPath.closeSubpath()
+    
+        let square2 = Path(cgPath: square2CGPath)
+        let result = square1.subtracting(square2)
+        
+        let expectedResultCGPath = CGMutablePath()
+        expectedResultCGPath.move(to: CGPoint.zero)
+        expectedResultCGPath.addLine(to: CGPoint(x: 1.0, y: 1.0))
+        expectedResultCGPath.addLine(to: CGPoint(x: 2.0, y: 0.0))
+        expectedResultCGPath.addLine(to: CGPoint(x: 2.0, y: 2.0))
+        expectedResultCGPath.addLine(to: CGPoint(x: 0.0, y: 2.0))
+        expectedResultCGPath.closeSubpath()
+        
+        let expectedResult = Path(cgPath: expectedResultCGPath)
+        
+        XCTAssertEqual(result.subpaths.count, expectedResult.subpaths.count)
+        XCTAssertTrue(componentsEqualAsideFromElementOrdering(result.subpaths[0], expectedResult.subpaths[0]))
+    }
+    
     func testCrossingsRemoved() {
         let points: [CGPoint] = [
             CGPoint(x: 0, y: 0),
@@ -456,4 +496,32 @@ class PathTests: XCTestCase {
         XCTAssertTrue(componentsEqualAsideFromElementOrdering(result.subpaths[0], square.subpaths[0]))
     }
     
+    func testCrossingsRemovedEdgeCase() {
+        // this is an edge cases which caused difficulty in practice
+        // the contour, which intersects at (1,1) creates two squares, one with -1 winding count
+        // the other with +1 winding count
+        // incorrect implementation of this algorithm previously interpretted
+        // the crossing as an entry / exit, which would completely cull off the square with +1 count
+        
+        let points = [CGPoint(x: 0, y: 1),
+                      CGPoint(x: 2, y: 1),
+                      CGPoint(x: 2, y: 2),
+                      CGPoint(x: 1, y: 2),
+                      CGPoint(x: 1, y: 0),
+                      CGPoint(x: 0, y: 0)]
+
+        let cgPath = CGMutablePath()
+        cgPath.addLines(between: points)
+        cgPath.closeSubpath()
+        
+        let contour = Path(cgPath: cgPath)
+        XCTAssertEqual(contour.windingCount(CGPoint(x: 0.5, y: 0.5)), -1) // winding count at center of one square region
+        XCTAssertEqual( contour.windingCount(CGPoint(x: 1.5, y: 1.5)), 1) // winding count at center of other square region
+
+        let crossingsRemoved = contour.crossingsRemoved()
+
+        XCTAssertEqual(crossingsRemoved.subpaths.count, 1)
+        XCTAssertTrue(componentsEqualAsideFromElementOrdering(crossingsRemoved.subpaths[0], contour.subpaths[0]))
+    }
+    
 }
diff --git a/BezierKit/Library/AugmentedGraph.swift b/BezierKit/Library/AugmentedGraph.swift
index 8f161952..ffa2e59b 100644
--- a/BezierKit/Library/AugmentedGraph.swift
+++ b/BezierKit/Library/AugmentedGraph.swift
@@ -75,11 +75,9 @@ internal class PathLinkedListRepresentation {
         
         var list = self.lists[location.componentIndex]
         
-        if location.t == 0 {
-            // this vertex needs to replace the start vertex of the element
-            insertIntersectionVertex(v, replacingVertexAtStartOfElementIndex: location.elementIndex, inList: &list)
-        }
-        else if location.t == 1 {
+        assert(location.t != 0, "intersects are assumed pre-processed to have a t=1 intersection at the previous path element instead!")
+        
+        if location.t == 1 {
             // this vertex needs to replace the end vertex of the element
             insertIntersectionVertex(v, replacingVertexAtStartOfElementIndex: Utils.mod(location.elementIndex+1, list.count), inList: &list)
         }
@@ -134,7 +132,8 @@ internal class PathLinkedListRepresentation {
     }
     
     fileprivate func markEntryExit(_ path: Path, _ nonCrossingComponents: inout [PathComponent], useRelativeWinding: Bool = false) {
-        let fillRule = PathFillRule.winding
+        let fillRule: PathFillRule = useRelativeWinding ? .winding : .evenOdd
+//        let fillRule = PathFillRule.winding
         for i in 0..<lists.count {
             
             // determine winding counts relative to the first vertex
@@ -158,6 +157,9 @@ internal class PathLinkedListRepresentation {
                 let cross = (side1 != side2)
                 
                 if cross {
+                    // TODO: there's an issue when corners intersect (try AugmentedGraphTests.testCornersIntersect which has this problem, even though it passes)
+                    // the relative winding count can be decremented both for entry and for exit. This is not an issue with the even-odd winding rule, but using
+                    // winding it can be an issue
                     let c = CGPoint.cross(v2, n2)
                     if c < 0 {
                         relativeWindingCount += 1
@@ -182,6 +184,22 @@ internal class PathLinkedListRepresentation {
                     }
                 }
                 initialWinding = -minimumWinding
+                
+                let prev = lists[i][0].emitPrevious()
+                let a = prev.compute(0.5)
+                // TODO: 1.0e-5 is a magic number (just an arbitrary small value)
+                let b = a + 1.0e-5 * prev.normal(0.5)
+                //let c = a - 1.0e-5 * prev.normal(0.5)
+                
+                let w1 = path.windingCount(b)
+                //let w2 = path.windingCount(c)
+                
+                // print("w1 = \(w1)")
+                // print("w2 = \(w2)")
+                
+                if w1 == initialWinding-1 {
+                    initialWinding = w1
+                }
             }
             else {
                 initialWinding = path.windingCount(lists[i][0].emitPrevious().compute(0.5))
@@ -202,9 +220,16 @@ internal class PathLinkedListRepresentation {
                 guard v.isIntersection else {
                     return
                 }
-                let wasInside = windingCountImpliesContainment(windingCount, using: fillRule)
+                var wasInside = windingCountImpliesContainment(windingCount, using: fillRule)
+                if useRelativeWinding {
+                    wasInside = windingCountImpliesContainment(windingCount, using: fillRule) && windingCountImpliesContainment(windingCount+1, using: fillRule)
+                }
                 windingCount = v.intersectionInfo.nextWinding
-                let isInside = windingCountImpliesContainment(windingCount, using: fillRule)
+                var isInside = windingCountImpliesContainment(windingCount, using: fillRule)  || (useRelativeWinding && windingCountImpliesContainment(windingCount+1, using: fillRule))
+                if useRelativeWinding {
+                    isInside = windingCountImpliesContainment(windingCount, using: fillRule) && windingCountImpliesContainment(windingCount+1, using: fillRule)
+                }
+
                 v.intersectionInfo.isEntry = wasInside == false && isInside == true
                 v.intersectionInfo.isExit = wasInside == true && isInside == false
                 if v.intersectionInfo.isEntry || v.intersectionInfo.isExit {
@@ -244,6 +269,7 @@ internal enum BooleanPathOperation {
     case union
     case difference
     case intersection
+    case removeCrossings
 }
 
 internal class AugmentedGraph {
@@ -293,6 +319,8 @@ internal class AugmentedGraph {
     
     private func shouldMoveForwards(fromVertex v: Vertex, forOperation operation: BooleanPathOperation, isOnFirstCurve: Bool) -> Bool {
         switch operation {
+        case .removeCrossings:
+            fallthrough
         case .union:
             return v.intersectionInfo.isExit
         case .difference:
@@ -309,6 +337,8 @@ internal class AugmentedGraph {
         }
         var pathComponents: [PathComponent] = []
         switch operation {
+        case .removeCrossings:
+            pathComponents += nonCrossingComponents1 // TODO: hmm7
         case .union:
             pathComponents += nonCrossingComponents1.filter { path2.contains(anyPointOnComponent($0)) == false }
             pathComponents += nonCrossingComponents2.filter { path1.contains(anyPointOnComponent($0)) == false }
diff --git a/BezierKit/Library/Path.swift b/BezierKit/Library/Path.swift
index 15622c9f..79644d66 100644
--- a/BezierKit/Library/Path.swift
+++ b/BezierKit/Library/Path.swift
@@ -45,6 +45,10 @@ internal func windingCountImpliesContainment(_ count: Int, using rule: PathFillR
         return mutablePath.copy()!
     }()
     
+    @objc public var isEmpty: Bool {
+        return self.subpaths.isEmpty // components are not allowed to be empty
+    }
+    
     public lazy var boundingBox: BoundingBox = {
         return self.subpaths.reduce(BoundingBox.empty) {
             BoundingBox(first: $0, second: $1.boundingBox)
@@ -186,6 +190,12 @@ internal func windingCountImpliesContainment(_ count: Int, using rule: PathFillR
         }
     }
     
+    @objc(offsetWithDistance:) public func offset(distance d: CGFloat) -> Path {
+        return Path(subpaths: self.subpaths.map {
+            $0.offset(distance: d)
+        })
+    }
+    
     @objc public func contains(_ point: CGPoint, using rule: PathFillRule = .winding) -> Bool {
         let count = self.windingCount(point)
         return windingCountImpliesContainment(count, using: rule)
@@ -202,6 +212,12 @@ internal func windingCountImpliesContainment(_ count: Int, using rule: PathFillR
     }
     
     @objc(unionedWithPath:threshold:) public func `union`(_ other: Path, threshold: CGFloat=BezierKit.defaultIntersectionThreshold) -> Path {
+        guard self.isEmpty == false else {
+            return other
+        }
+        guard other.isEmpty == false else {
+            return self
+        }
         return self.performBooleanOperation(.union, withPath: other, threshold: threshold)
     }
     
@@ -210,25 +226,23 @@ internal func windingCountImpliesContainment(_ count: Int, using rule: PathFillR
     }
     
     @objc(crossingsRemovedWithThreshold:) public func crossingsRemoved(threshold: CGFloat=BezierKit.defaultIntersectionThreshold) -> Path {
-        assert(self.subpaths.count <= 1, "todo: support multi-component paths")
-        guard self.subpaths.count > 0 else {
-            return Path()
-        }
-        let component = self.subpaths[0]
-        let intersections = component.intersects(threshold: threshold).compactMap { (i: PathComponentIntersection) -> PathIntersection? in
-            guard i.indexedComponentLocation1.elementIndex <= i.indexedComponentLocation2.elementIndex else {
-                return nil
+        //        assert(self.subpaths.count <= 1, "todo: support multi-component paths")
+        return self.subpaths.reduce(Path(), { result, component  in
+            // TODO: this won't work properly if components intersect
+            let intersections = component.intersects(threshold: threshold).map {(i: PathComponentIntersection) -> PathIntersection in
+                return PathIntersection(indexedPathLocation1: IndexedPathLocation(componentIndex: 0, elementIndex: i.indexedComponentLocation1.elementIndex, t: i.indexedComponentLocation1.t),
+                                        indexedPathLocation2: IndexedPathLocation(componentIndex: 0, elementIndex: i.indexedComponentLocation2.elementIndex, t: i.indexedComponentLocation2.t))
             }
-            return PathIntersection(indexedPathLocation1: IndexedPathLocation(componentIndex: 0, elementIndex: i.indexedComponentLocation1.elementIndex, t: i.indexedComponentLocation1.t),
-                                    indexedPathLocation2: IndexedPathLocation(componentIndex: 0, elementIndex: i.indexedComponentLocation2.elementIndex, t: i.indexedComponentLocation2.t))
-        }
-        if intersections.count == 0 {
-            return self
-        }
-        let augmentedGraph = AugmentedGraph(path1: self, path2: self, intersections: intersections)
-        return augmentedGraph.booleanOperation(.union)
+            guard intersections.isEmpty == false else {
+                return Path(subpaths: result.subpaths + [component])
+            }
+            let singleComponentPath = Path(subpaths: [component])
+            let augmentedGraph = AugmentedGraph(path1: singleComponentPath, path2: singleComponentPath, intersections: intersections)
+            let path = augmentedGraph.booleanOperation(.removeCrossings)
+            return Path(subpaths: result.subpaths + path.subpaths)
+        })
     }
-    
+
     @objc public func disjointSubpaths() -> [Path] {
         
         var paths: Set<Path> = Set<Path>()
diff --git a/BezierKit/Library/PathComponent.swift b/BezierKit/Library/PathComponent.swift
index 552d50f3..b5ec059f 100644
--- a/BezierKit/Library/PathComponent.swift
+++ b/BezierKit/Library/PathComponent.swift
@@ -50,6 +50,7 @@ public final class PathComponent: NSObject, NSCoding {
     }()
     
     public init(curves: [BezierCurve]) {
+        precondition(curves.isEmpty == false, "Path components are by definition non-empty.")
         self.curves = curves
     }
     
@@ -92,9 +93,13 @@ public final class PathComponent: NSObject, NSCoding {
             let c1 = o1 as! BezierCurve
             let c2 = o2 as! BezierCurve
             let elementIntersections = c1.intersects(curve: c2, threshold: threshold)
-            let pathComponentIntersections = elementIntersections.map { (i: Intersection) -> PathComponentIntersection in
+            let pathComponentIntersections = elementIntersections.compactMap { (i: Intersection) -> PathComponentIntersection? in
                 let i1 = IndexedPathComponentLocation(elementIndex: i1, t: i.t1)
                 let i2 = IndexedPathComponentLocation(elementIndex: i2, t: i.t2)
+                guard i1.t != 0.0 && i2.t != 0.0 else {
+                    // we'll get this intersection at t=1 on the neighboring path element(s) instead
+                    return nil
+                }
                 return PathComponentIntersection(indexedComponentLocation1: i1, indexedComponentLocation2: i2)
             }
             intersections += pathComponentIntersections
@@ -114,7 +119,7 @@ public final class PathComponent: NSObject, NSCoding {
                     elementIntersections = c.intersects(threshold: threshold)
                 }
             }
-            else {
+            else if i1 < i2 {
                 // we are intersecting two distinct path elements
                 elementIntersections = c1.intersects(curve: c2, threshold: threshold).filter {
                     if i1 == Utils.mod(i2+1, self.curves.count) && $0.t1 == 0.0 {
diff --git a/BezierKit/Library/Utils.swift b/BezierKit/Library/Utils.swift
index cdc2e2eb..8960944d 100644
--- a/BezierKit/Library/Utils.swift
+++ b/BezierKit/Library/Utils.swift
@@ -382,6 +382,12 @@ internal class Utils {
     static func pairiteration<C1, C2>(_ c1: Subcurve<C1>, _ c2: Subcurve<C2>, _ results: inout [Intersection], _ threshold: CGFloat = BezierKit.defaultIntersectionThreshold) {
         let c1b = c1.curve.boundingBox
         let c2b = c2.curve.boundingBox
+        
+        if results.count > 20 {
+            // TODO: better bailout conditions
+            return
+        }
+        
         if c1b.overlaps(c2b) == false {
             return
         }
diff --git a/BezierKit/MacDemos/Demos.swift b/BezierKit/MacDemos/Demos.swift
index f04fbb2d..3567016e 100644
--- a/BezierKit/MacDemos/Demos.swift
+++ b/BezierKit/MacDemos/Demos.swift
@@ -433,6 +433,10 @@ class Demos {
                                 if let mouse = demoState.lastInputLocation {
                                     
                                     //let m2 = CGPoint(x: -21.19140625, y: 131.38671875)
+                                    //let me2 = CGPoint(x: 24.34375, y: 110.703125)
+                                    
+                                    //let me3 = CGPoint(x: -7.78515625, y: 161.7265625) // seems to cause an issue because intersections[5].t = 0.99999422905833845, clamping the t values when they are appropximately 1 or 0 seems to work (but fix not applied)
+                                    // let me4 = CGPoint(x: 22.41796875, y: 168.48046875) // caused an infinite loop or graphical glitches
                                     
                                     var translation = CGAffineTransform.init(translationX: mouse.x, y: mouse.y)
                                     let cgPath2: CGPath = CTFontCreatePathForGlyph(font, glyph2, &translation)!